Problems on Finite Automata and the Exponential Time Hypothesis

Fernau, Henning; Krebs, Andreas

doi:10.3390/a10010024

Cited by 23 publications

(15 citation statements)

References 44 publications

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“…The DCFS proceedings has a slightly worse bound of O(m4 • |Σ|), and the specifics of the improved version will appear in a future paper 2. This observation, with a justification different from the below one, was suggested by an anonymous referee.…”

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confidence: 88%

“…An m-state AFA A with acceptance width k can be simulated by an NFA where the states are k-tuples of states of A and transitions of the NFA simulate at most k parallel computations of A. It is known that the emptiness problem for NFAs can be solved in linear time, with respect to the number of states, using a breadth first search [4]. The transformation from Lemma 1 then yields a polynomial-time algorithm to decide emptiness for a finite acceptance width AFA.…”

Section: Decision Problems For Pared Tree Width and Acceptance Widthmentioning

confidence: 99%

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Alternating Finite Automata with Limited Universal Branching

Keeler

Salomaa

2020

Language and Automata Theory and Applications

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We consider measures that limit universal parallelism in computations of an alternating finite automaton (AFA). Maximum pared tree width counts the largest number of universal branches in any computation and acceptance width counts the number of universal branches in the best accepting computation, i.e., in the accepting computation with least universal parallelism. We give algorithms to decide whether the maximum pared tree width or the acceptance width of an AFA are bounded by an integer k. For a constant k the algorithm for maximum pared tree width operates in polynomial time. An AFA with m states and acceptance width k can be converted to an NFA with (m + 1) k states. We consider corresponding lower bounds for the transformation. The tree width of an AFA counts the number of all (existential and universal) branches of the computation. We give upper and lower bounds for converting an AFA of bounded tree width to a DFA.

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confidence: 88%

Section: Decision Problems For Pared Tree Width and Acceptance Widthmentioning

confidence: 99%

Alternating Finite Automata with Limited Universal Branching

Keeler

Salomaa

2020

Language and Automata Theory and Applications

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“…Potechin and Shallit [2020] show a reduction from Orthogonal Vectors to the acceptance problem for (a subclass of) NFA and a reduction from triangle finding to (unary) NFA acceptance. Fernau and Krebs [2017] establish conditional lower bounds for a variety of automata-theoretic problems beyond P. Wehar and co-authors have shown that faster algorithms for various intersection non-emptiness problems have consequences for structural complexity classes [de Oliveira Oliveira and Wehar 2020; Swernofsky and Wehar 2015;Wehar 2014].…”

Section: Related Workmentioning

confidence: 99%

Subcubic certificates for CFL reachability

Chistikov

Majumdar

Schepper

2022

Proc. ACM Program. Lang.

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Many problems in interprocedural program analysis can be modeled as the context-free language (CFL) reachability problem on graphs and can be solved in cubic time. Despite years of efforts, there are no known truly sub-cubic algorithms for this problem. We study the related certification task: given an instance of CFL reachability, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates ( O ( n 2 ) in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. The certificates are based on grammar-based compression of paths (for reachability) and on invariants represented as matrix inequalities (for non-reachability). Thus, CFL reachability lies in nondeterministic and co-nondeterministic subcubic time. A natural question is whether faster algorithms for CFL reachability will lead to faster algorithms for combinatorial problems such as Boolean satisfiability (SAT). As a consequence of our certification results, we show that there cannot be a fine-grained reduction from SAT to CFL reachability for a conditional lower bound stronger than n ω , unless the nondeterministic strong exponential time hypothesis (NSETH) fails. In a nutshell, reductions from SAT are unlikely to explain the cubic bottleneck for CFL reachability. Our results extend to related subcubic equivalent problems: pushdown reachability and 2NPDA recognition; as well as to all-pairs CFL reachability. For example, we describe succinct certificates for pushdown non-reachability (inductive invariants) and observe that they can be checked in matrix multiplication time. We also extract a new hardest 2NPDA language, capturing the “hard core” of all these problems.

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“…We will also need the next problem from [11] which is PSPACE-complete in general, but NP-complete for unary automata, see [5].…”

Section: Appendixmentioning

confidence: 99%

Computational Complexity of Synchronization under Regular Commutative Constraints

Hoffmann

2020

Preprint

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Here we study the computational complexity of the constrained synchronization problem for the class of regular commutative constraint languages. Utilizing a vector representation of regular commutative constraint languages, we give a full classification of the computational complexity of the constraint synchronization problem. Depending on the constraint language, our problem becomes PSPACE-complete, NPcomplete or polynomial time solvable. In addition, we derive a polynomial time decision procedure for the complexity of the constraint synchronization problem, given some constraint automaton accepting a commutative language as input.

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Problems on Finite Automata and the Exponential Time Hypothesis

Cited by 23 publications

References 44 publications

Alternating Finite Automata with Limited Universal Branching

Alternating Finite Automata with Limited Universal Branching

Subcubic certificates for CFL reachability

Computational Complexity of Synchronization under Regular Commutative Constraints

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